# Summer Term 2018

Zhengfang Wang (BICMR, Beijing). Homotopy algebra structures on Tate-Hochschild cohomology. Friday 4th May, 1:30-2:30pm. Huxley 144.

Abstract: The Tate-Hochschild cohomology of a singular space X is defined as the graded endomorphism ring of the diagonal inside the singularity category of X x X. Singularity categories are introduced by Buchweitz and independently by Orlov, which have played a central role in noncommutative geometry and homological mirror symmetry.
In this talk, we construct an explicit complex to compute the Tate-Hochschild cohomology. We prove that this complex is an algebra over the little 2-discs operad, namely, the Deligne conjecture for this complex holds. We will also talk about a joint work with M. Rivera that the Tate-Hochschild cohomology of a simply-connected closed manifold recovers the Rabinowitz-Floer homology of the unit disc cotangent bundle.

Dougal Davis (King’s College). The Grothendieck-Springer resolution for the stack of principal bundles on an elliptic curve. Friday 11th May, 1:30-2:30pm. Huxley 144.

Abstract: If G is a simply connected simple algebraic group, the associated adjoint quotient is the GIT quotient of the Lie algebra by the adjoint action of G. This is very far from being a genuine quotient; in fact, the morphism from the Lie algebra to the adjoint quotient is a flat family of affine varieties with many singular fibres. After pulling back along a Weyl group covering of the base, this family has a natural simultaneous resolution of singularities, called the Grothendieck-Springer resolution. As a very basic application, one can use the Grothendieck-Springer resolution to show that the codimension 2 singularities of the most singular fibre of the adjoint quotient map are du Val of the same type as G.
In this talk, I will explain a closely related picture for the stack of principal G-bundles on an elliptic curve. In this context, the analogue of the adjoint quotient map is a natural extension to the stack of all G-bundles of the morphism from the stack of semistable G-bundles to its coarse moduli space. I will introduce an analogue of the Grothendieck-Springer resolution for this map and use it to describe the codimension 2 singularities of the locus of unstable bundles in some examples.

Yang-Hui He (City University of London). Learning Algebraic Geometry: Lessons from the String Landscape. Friday 18th May, 1:30-2:30pm. Huxley 139.

Abstract: We propose a paradigm to machine-learn the ever-expanding databases which have emerged in mathematical physics, algebraic geometry and particle phenomenology, as diverse as the statistics of string vacua and the classification of varieties.
As concrete examples, we establish multi-layer neural networks as both classifiers and predictors and train them with a host of available data ranging from Calabi-Yau manifolds to quiver representations for gauge theories, achieving impressive precision in a matter of minutes.
This paradigm should prove useful in various investigations in landscapes in physics as well as pure mathematics.

Matteo Ruggiero (Université Paris 7). Links of normal surface singularities and non-Kahler manifolds. Friday 25th May, 1:30-2:30pm. BLKT 112 – Lecture Theatre 3.

Abstract: As a first example of non-Kahler surfaces, Hopf surfaces can be constructed as the space of orbits of contracting automorphisms of the affine plane fixing the origin. This construction can be generalized to contracting germs of topological degree one, that are not necessarily automorphisms. The compactification of the space of orbits of such local dynamical systems gives other non-Kahler surfaces, called Kato surfaces. In a joint work with Lorenzo Fantini and Charles Favre, we investigate analogous constructions for normal surface singularities. In particular, we classify normal surface singularities whose link can be embedded in a compact complex surface without disconnecting it.

No seminar on Friday 1st June due to this event.

Olivier Schiffmann (Université de Paris-Sud). Cohomological Hall algebras associated to curves. Friday 8th June, 1:30-2:30pm. Huxley 139.

Abstract: To any compact Riemann surface one can associate (at least) two kinds of cohomological Hall algebras: one whose underlying vector space is the (singular) cohomology of the moduli stack of vector bundles (or coherent sheaves) on X, and one whose underlying vector space is the (singular) cohomology of the moduli stack of Higgs bundles (or Higgs sheaves) on X. We will describe these algebras and give some applications and conjectures.

Please note there will be two seminars on 15th June:
Thomas Prince (University of Oxford). Real Lagrangians from toric degenerations of the quintic threefold. Friday 15th June, 1:30-2:30pm. Huxley 139.

Abstract: Given a toric degeneration of a Calabi–Yau threefold one may construct an integral affine manifold with singularities, the candidate base of an SYZ fibration, explored in detail in work of Gross, Haase–Zharkov, and Ruan. The integral affine structure itself determines a topological (in fact symplectic) model for the SYZ fibration, and moreover there is a canonical involution on this torus fibration, first studied by Castano-Bernard–Matessi. We study the induced real Lagrangian L in the case of the quintic in detail. In particular we show that passing between toric degenerations induces a sequence of half-integral Dehn surgeries, and that the first cohomology of L, with Z2 coefficients, has dimension 101, independently of the choice of degeneration. Finally we interpret the image of a map between (affine) Hodge groups of degree three defined by Castano-Bernard–Matessi as a (mod 2) tropical cycle. This is joint work with Hülya Arguz.

Fedor Bogomolov (New York University). $PGL(2)$-invariants of collections of torsion points of elliptic curves. Friday 15th June, 3:00-4:00pm. Huxley 658.

Abstract: The main object of the talk is a (complex) elliptic curve $E$ with a standard degree $2$ projection $\pi$ on $P^1$. Assuming that we fix one of the ramification points as a zero we obtain a subset $PE_{tors}$ of the images of torsion points on $E$ inside $P^1$. These sets are different and we have shown jointly with Yuri Tschinkel that these sets are very different for different elliptic curves- they have finite intersection for any two nonisomorphic elliptic curves. However some subsets of the above $PE_{tors}$ are $PGL(2)$ equivalent. This holds for the images of points of order $3$ and order $4$
In this talk I am going to discuss generlal problem of the behavior of $PGL(2)$-invariants of the $k$-tuples of the images of torsion points of different order.
For every subset of different $k$ points in $P^1$ we can define it’s image in the moduli $M_{0,k}$ of $k$-tuples of points which is essentially a quotient of projective space $S^kP^1= P^k$ by the action of $PGL(2)$. Thus $M_{0,k}$ is a rational variety of dimension $k-3$. If we consider the images of points of finite order in different elliptic curves under natural projection onto $P^1$ then we obtain an( infinite) system of modular type curves with maps into $M_{0,k}$ I will formulate three conjectures( semi theorems) about properties of such maps which provide a possiblity of realistic universal picture for intersections between subset of $PE^i_{tors},i=1,2$ for different elliptic curves $E^i$
These conjectures are formulated in our joint article with Yuri Tcshinkel and Hang Fu. Note that there are pairs of curves $E^i$ with big intersection $\geq 22$ of $PE^i_{tors},i=1,2$ as it was shown in our joint article.

Fabian Haiden (Harvard University). Spectral networks and stability conditions. Friday 22nd June, 1:30-2:30pm. Huxley 139.

Abstract: Spectral networks were introduced in theoretical physics in order to count BPS states. They are graphs with additional algebraic data drawn on a surface and minimizing a weighted total length. Heuristically, they are degenerations of higher dimensional special Lagrangian submanifolds. The main mathematical conjecture about them is that they correspond to stable objects of a stability condition on a Fukaya-type category. I will illustrate how this works in examples related to Dynkin quivers and stacky projective lines. This is an ongoing joint project with L. Katzarkov, M. Kontsevich, P. Pandit, and C. Simpson.